On Higher Dimensional Point Sets in General Position

A finite point set in ?^d is in general position if no d + 1 points lie on a common hyperplane. Let ?_d(N) be the largest integer such that any set of N points in ?^d with no d + 2 members on a common hyperplane, contains a subset of size ?_d(N) in general position. Using the method of hypergraph containers, Balogh and Solymosi showed that ??(N) < N^{5/6 + o(1)}. In this paper, we also use the container method to obtain new upper bounds for ?_d(N) when d ? 3. More precisely, we show that if d is odd, then ?_d(N) < N^{1/2 + 1/(2d) + o(1)}, and if d is even, we have ?_d(N) < N^{1/2 + 1/(d-1) + o(1)}. We also study the classical problem of determining the maximum number a(d,k,n) of points selected from the grid [n]^d such that no k + 2 members lie on a k-flat. For fixed d and k, we show that a(d,k,n)? O(n^{d/{2?(k+2)/4?}(1- 1/{2?(k+2)/4?d+1})}), which improves the previously best known bound of O(n^{d/?(k + 2)/2?}) due to Lefmann when k+2 is congruent to 0 or 1 mod 4